The way forward

It is a new year. Time to look ahead, having completed a project toward the end of last year. Well, it still has some things I can look at, but I did make a bit of a breakthrough (if removing an error that resolved an annoying divergence can be called a “breakthrough”). Now, it is natural to look further ahead and ask oneself where one is heading.

In my case, I still hope to develop a formalism that is powerful enough to formulate fundamental theories that incorporate the dynamics of the standard model with gravity. But wait, isn’t that what they are trying to do with string theory and all those other theories?

No, there is a difference. The idea is not to build the speculative aspects of a new theory into the formalism itself. It seems to me that all the currently popular attempts to formulate theories of fundamental physics incorporate speculative ideas into the mathematics of the formalism itself. If they fail, the whole thing fails and there is nothing to salvage.

The only physics that should be built into the formalism is physics that has been established as scientific knowledge. That is the situation with quantum field theory. It has special relativity built into it, because that has been confirmed experimentally. Thus it allows speculative new theories to be formulated.

The inclusion of special relativity may also be the reason why quantum field theory cannot model gravity, which goes beyond special relativity. The obvious thing is then to modify the part that involves the special relativity and to replace it with general relativity. Well that has been tried and did not work.

I think the reason why the obvious extension of quantum field theory to incorporate gravity did not work is because it does not incorporate the formulation of states. Gravity depends on the nature of states. Therefore, my idea is to replace the path integral formulation with a functional phase space. Such a functional phase space formulation allows the definition of arbitrary complicated states. Such a functional phase space formulation is an idea that has been bounced around in the literature, but I have not seen a complete formulation that can handle gravity.

The origin of Heisenberg uncertainty

Demystifying quantum mechanics II

Perhaps the one thing that everyone thinks about when they hear talk about quantum mechanics is Heisenberg’s uncertainty principle. It may even sometimes be considered as the essence of quantum mechanics. Now what would you say if I tell you that the Heisenberg uncertainty principle is not a fundamental principle and that the origin of this principle is not found in quantum mechanics? The fundamental origin of this uncertainty is a purely mathematical property and the reason that quantum mechanics inherited this principle is simply a result of the Planck relationship.

Werner Heisenberg

I have discussed this issue to some extent before. However, it forms an important part of the knowledge that would help to demystify quantum mechanics. Therefore, it deserves more attention.

Before the advent of quantum mechanics, the state of a particle was considered to be completely described by its position and velocity (momentum). The dynamics of a system could then be represented by a diagram showing position and velocity of the particle as a function of time. For historical reasons, the domain of such a diagram is called phase space. For a one-dimensional system (such as a harmonic oscillator), it would give a two-dimensional graph with position on one axis and velocity on the other. The state of the system is a point on the two-dimensional plane that moves along some trajectory as a function of time. For a harmonic oscillator, this trajectory is a circle.

A mathematical property (in Fourier analysis), which may have seemed to be complete unrelated at the time, is that the width of the spectrum of a function has a lower limit that is proportional to the inverse of the width of the function. This property has nothing to do with physical reality. It is a purely logical fact that can be proven with the aid of mathematics. If the function is, for instance, interpreted as the probability distribution of the position of a particle, the width of the function would represent the uncertainty in its location.

This mathematical uncertainty property was transferred to phase space by Planck’s relation, which links the independent variable of the spectrum (the wave number) with the momentum or velocity of the particle. The implication is that one cannot represent the state of a quantum particle with a single dimensionless point on phase space in quantum mechanics. Hence, the Heisenberg uncertainty principle.

So, the uncertainty associated with Heisenberg’s principle is inevitable due to Planck’s relation. And it is founded on pure logic in terms of which mathematics is based. Planck’s relation is the only physics that enters the picture. The Heisenberg uncertainty principle is therefore not a separate principle that is independent of Planck’s relationship as far as the physics is concerned.

Now, there are a few subtleties that we can address. There are also some interesting consequences based in this understanding, but I’ll leave these for later.